Properties

Degree 1
Conductor 349
Sign $0.983 - 0.179i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.619 − 0.785i)2-s + (−0.0901 − 0.995i)3-s + (−0.232 + 0.972i)4-s + (0.750 + 0.661i)5-s + (−0.725 + 0.687i)6-s + (0.700 + 0.713i)7-s + (0.907 − 0.419i)8-s + (−0.983 + 0.179i)9-s + (0.0541 − 0.998i)10-s + (−0.994 − 0.108i)11-s + (0.989 + 0.143i)12-s + (−0.922 + 0.386i)13-s + (0.126 − 0.992i)14-s + (0.590 − 0.806i)15-s + (−0.891 − 0.452i)16-s + (0.907 + 0.419i)17-s + ⋯
L(s,χ)  = 1  + (−0.619 − 0.785i)2-s + (−0.0901 − 0.995i)3-s + (−0.232 + 0.972i)4-s + (0.750 + 0.661i)5-s + (−0.725 + 0.687i)6-s + (0.700 + 0.713i)7-s + (0.907 − 0.419i)8-s + (−0.983 + 0.179i)9-s + (0.0541 − 0.998i)10-s + (−0.994 − 0.108i)11-s + (0.989 + 0.143i)12-s + (−0.922 + 0.386i)13-s + (0.126 − 0.992i)14-s + (0.590 − 0.806i)15-s + (−0.891 − 0.452i)16-s + (0.907 + 0.419i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.983 - 0.179i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.983 - 0.179i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(349\)
\( \varepsilon \)  =  $0.983 - 0.179i$
motivic weight  =  \(0\)
character  :  $\chi_{349} (85, \cdot )$
Sato-Tate  :  $\mu(87)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 349,\ (0:\ ),\ 0.983 - 0.179i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9207140565 - 0.08334892977i$
$L(\frac12,\chi)$  $\approx$  $0.9207140565 - 0.08334892977i$
$L(\chi,1)$  $\approx$  0.8033886430 - 0.2265058849i
$L(1,\chi)$  $\approx$  0.8033886430 - 0.2265058849i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−24.96962002357272708496734357992, −24.10024586355415406965603458939, −23.29810957054908337997166534866, −22.27470388387726203364857901634, −21.20101369616333569371761624331, −20.41859815216850681307920815647, −19.80554016206253448097701512501, −18.04833687452342805883609867467, −17.75189710446992367121192323520, −16.50145116754160870638041906726, −16.37559868654718092098276011897, −15.06450732800668046683939887667, −14.28063194687219965613097307362, −13.519460402122816123879963339409, −11.98186279129744440909623946728, −10.565883254064870890114486668332, −10.08914184511913358506947763407, −9.29075769400061020811207336136, −8.15433498788395137284672172603, −7.40137233644359212653381345069, −5.70076860886156656580498365208, −5.213856128617637083478333557371, −4.32947289324762840458424619510, −2.43355818481180454972690630335, −0.7603307159415483122206392946, 1.41320436785362308701744936555, 2.33296150446648183617943812802, 3.043117325576308039029371780042, 5.04698691597444661378281595340, 6.058839618198614114239338545247, 7.48970043745102428769749332602, 7.94216367296324059303279594925, 9.22769636750414426056562634142, 10.175705461681545156739036289558, 11.18736378291968492496564625017, 12.02291710889459182106399143889, 12.7763377071455751929204640276, 13.8845648958792666636904434920, 14.53779968574782775408294541894, 16.13639157028661207657729847503, 17.36517000163247866901777127248, 17.93306893949622260513671285987, 18.531390879059874967294824957492, 19.22183390599180344685981159285, 20.29058677268907235237819104223, 21.386727723814208427629386260616, 21.85803966426253884234700237618, 22.94057589024699151420541716513, 24.06455986016104813238061782312, 24.9740816168749061453265023323

Graph of the $Z$-function along the critical line